Matrix Rank
Find the rank by row reduction · 행렬 계수(rank)
1. What Is the Rank of a Matrix?
The rank of a matrix is the number of linearly independent rows — equivalently, the number of linearly independent columns (the two counts are always equal). Intuitively, it measures how much genuine information the matrix carries: how many of its rows are not just combinations of the others.
A matrix has full rank when its rank equals the smaller of its row and column counts. Rank ties together invertibility, the dimension of the column space, and — crucially — whether a linear system has a solution and how many.
행렬의 계수(rank)는 일차독립인 행의 개수이며, 일차독립인 열의 개수와 항상 같습니다. 행렬이 담은 실제 정보량(다른 행들의 조합이 아닌 행이 몇 개인지)을 나타냅니다. 행과 열 수 중 작은 값과 계수가 같으면 풀계수(full rank)입니다.
2. How to Find Rank: Row Reduction
The reliable method is row reduction:
- Use elementary row operations to reduce the matrix to row-echelon form (each leading entry sits to the right of the one above it).
- Count the non-zero rows. That count — the number of pivots — is the rank.
Rows that reduce entirely to zeros were linear combinations of the others; they add nothing to the rank.
계수는 행 줄이기로 구합니다: ① 기본 행 연산으로 행 사다리꼴(row-echelon form)로 만들고, ② 0이 아닌 행(피벗)의 개수를 셉니다. 그 수가 계수입니다. 모두 0이 되는 행은 다른 행들의 일차결합이므로 계수에 기여하지 않습니다.
3. Worked 3×3 Example
Find the rank of A = [[1, 2, 3], [2, 4, 6], [1, 1, 1]].
- R₂ − 2R₁: [2, 4, 6] − 2[1, 2, 3] = [0, 0, 0] — the second row was just twice the first.
- R₃ − R₁: [1, 1, 1] − [1, 2, 3] = [0, −1, −2].
The echelon form is [[1, 2, 3], [0, −1, −2], [0, 0, 0]]. There are 2 non-zero rows, so rank(A) = 2. The matrix is 3×3 but not full rank — its rows live in a 2-dimensional space, and its determinant is therefore 0.
A = [[1, 2, 3], [2, 4, 6], [1, 1, 1]]에서 R₂ − 2R₁ = [0, 0, 0], R₃ − R₁ = [0, −1, −2]. 사다리꼴 [[1, 2, 3], [0, −1, −2], [0, 0, 0]]에서 0이 아닌 행이 2개이므로 rank(A) = 2 입니다. 풀계수가 아니므로 det(A) = 0 입니다.
4. What Rank Tells You
- Invertibility. An n×n matrix is invertible if and only if it has full rank n (equivalently det ≠ 0).
- Solution type (Rouché–Capelli). For a system Ax = b with n unknowns, compare rank(A) with rank of the augmented matrix [A | b]:
- rank(A) = rank([A | b]) = n → unique solution.
- rank(A) = rank([A | b]) < n → infinitely many solutions.
- rank(A) < rank([A | b]) → no solution.
- Column space. The rank is the dimension of the space spanned by the columns.
n×n 행렬은 풀계수 n일 때만 가역입니다(= det ≠ 0). 연립방정식 Ax = b(미지수 n개)는 rank(A)와 rank([A | b])를 비교합니다: 둘이 같고 = n이면 유일해, 둘이 같고 < n이면 무수히 많은 해, rank(A) < rank([A | b])이면 해가 없습니다(루셰–카펠리 정리). 계수는 열공간의 차원이기도 합니다.
5. Frequently Asked Questions
How do you calculate the rank of a matrix? Row-reduce it to echelon form and count the non-zero rows (the pivots). That count is the rank.
What does full rank mean? The rank equals the smaller of the number of rows and columns — no row or column is redundant. A full-rank square matrix is invertible.
How does rank relate to solutions of a system? Compare rank(A) to rank([A | b]): equal and equal to the unknown count means a unique solution; equal but smaller means infinitely many; unequal means none.
행 줄이기로 사다리꼴을 만들고 0이 아닌 행을 세면 계수입니다. 풀계수는 행·열 수 중 작은 값과 같다는 뜻이며, 풀계수 정사각 행렬은 가역입니다. rank(A)와 rank([A | b])를 비교해 해의 종류를 판별합니다.
Ready to practice? Drill rank and the rest of linear algebra on C:Matrix, or review the full matrix reference and the related linear systems and determinant.